# The sum of a man's age and his son is 66. What are their ages if the digits are reveresed ? The teacher said there are 3 answers.

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Let the age of the father is presented by the 2 digit number xy.

Then, the son's age is the 2 digit number yx.

==> The fathers age = 10x + y

==. The son's age = 10y + x

But we know that the sum of their ages is 66.

==> 10x + y + 10y + x = 66

==> 11x + 11y = 66

We will divide by 11

==> x + y = 6

Then the sum of the digit must be 6.

Let us determine the positive integers whose sums are 6.

==> 0 + 6 = 6

Then the fathers age = 60

and the son's age = 06

Also, : 2 + 4 = 6

==> The father's age = 42

The son's age = 24

Also, 5 + 1 = 6

==> The fathers age = 51

The son's age = 15

Also, 3 + 3 = 6

==> The father's age = 33

The son's age = 33

But this solution is impossible because the father's and son's age can not be the same.

Then, possible answers are:

**( 51 and 15 ) (24 and 42) and ( 60 and 6)**

Let the man's age be ab and the son's age be ba.

Now we are given that the sum of their ages is 66.

So ab + ba = 66

=> 10a + b + 10b + a = 66

=> 11a + 11b = 66

=> a + b = 6

If a = 6 , b = 0

If a = 5, b = 1

If a = 4, b = 2

For values of a = 3 , 2 , 1 and 0 the son's age is no longer less than the man.

Therefore the age of the man and the son can be

**(60, 6), (51 ,15) and (42 , 24)**

Let 10x+y be the fathers age.

The sun's age is in reverse digits by data. So the sun's age = 10y+x.

Naturally father's age is more than sun's age.

=> 10x+y > 10y +x.

=> 10x+y-10y-x > 0

=> 9x-9y > 0.

=> 9(x-y) > 0.

=> x-y > 0.

x > y.....(1)

Also the sum of their ages = (10x+y)+(10y+x) = 11(x+y) which is 66 by data.

So 11(x+y) = 66.

We divide both sides of 11(x+y) = 66 by 11:

x+y = 6.....(2)

So x> y and x+y = 6. So we have the choice: x= 3, x= 4, x = 5, x = 6.

Then the corresponding y values are y= 3, y = 2, y = 1 and y = 0.

So father's age = 33 , 42, 51, 60.

Sun's age : 33 , 24, 15, 06.

We exclude 33 as it is not practical.

**So father and sun's age are one of the pairs: (42 , 24) , (51, 15) , (60, 06).**